Jian-Huang She- A Matrix Model for Misner Universe and Closed String Tachyons

8/3/2019 Jian-Huang She- A Matrix Model for Misner Universe and Closed String Tachyons

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A Matrix Model for Misner Universeand Closed String Tachyons

Jian-Huang She

Institute of Theoretical Physics, Chinese Academy of Science,

P.O.Box 2735, Beijing 100080, P.R. China

Graduate School of the Chinese Academy of Sciences, Beijing 100080, P.R. China

We use D-instantons to probe the geometry of Misner universe, and calculate the

world volume field theory action, which is of the 1+0 dimensional form and highly non-

local. Turning on closed string tachyons, we see from the deformed moduli space of the

D-instantons that the spacelike singularity is removed and the region near the singularity

becomes a fuzzy cone, where space and time do not commute. When realized cosmologi-

cally there can be controllable trans-planckian effects. And the infinite past is now causally

connected with the infinite future, thus also providing a model for big crunch/big bang

transition. In the spirit of IKKT matrix theory, we propose that the D-instanton action

here provides a holographic description for Misner universe and time is generated dynam-

ically. In addition we show that winding string production from the vacua and instability

of D-branes have simple uniform interpretations in this second quantized formalism.

Sep. 2005

Emails: [email protected]



1. Introduction

The resolution of spacelike singularities is one of the most outstanding problems in the

study of quantum gravity. These singularities make appearance in many black holes and

cosmological models. Unfortunately it is very hard to get much information about them ingeneral situations. So in order to make progress on this issue, more controllable toy models

are proposed, the simplest of which may be the two dimensional Misner space, which can

be defined as the quotient of two dimensional Minkowski space by a boost transformation.

Nowadays string theory is widely regarded as the most promising candidate for a

quantum theory of gravity. And actually string theory does provide resolution for some

singularities, such as orbifolds[1], conifolds[2] and enhancons[3]. For spacelike singularities,

less has been achieved. For example, even the most familiar GR singularity inside the

Schwarzschild black hole has not yet been understood.

Misner space can be embedded into string theory by adding 8 additional flat directions,

and it is an exact solution of string theory at least at tree-level [ 4]. The dynamics of

particles and strings in Misner universe were much explored in the literature (see for

example [5] [6] [7] [8] [9], for a good review see [10]). In particular, it was realized in the

above papers that winding strings are pair-produced and they backreact on the geometry.

Hence they may play important role in the resolution of the singularity. Unfortunately, it

is fair to say that we still lack a sensible treatment of the backreactions.

Along another line, in the study of closed string tachyons[11] [12] [13], Misner space

has reemerged as a valuable model[14]. By imposing anti-periodic boundary conditions for

fermions on the spatial circle, one can get winding tachyons near the singularity which can

significantly deform the original geometry. It is argued [14] that the spacetime near the

spacelike singularity will be replaced by a new phase of the tachyon condensate. In their

case the influence of the winding modes to the spacetime geometry is more significant and

more tractable. It is mainly this work [14] that motivates our following study.

We will use D-branes to probe the background geometry. D-branes are attractivehere because they can feel distances smaller than string scale [15]. For Misner universe,

the singularity is localized in sub-string region in the time direction, so we will use D-

instantons as probes. Recently, D0 and D1-branes in Misner space were studied in [16],

and it was found that they are both unstable due to open string pair production and closed

string emission.

1



In fact, the usefulness of D-instantons has deeper reasons. It was conjectured in [17]

that the large N limit of the supersymmetric matrix quantum mechanics describing D0-

branes provides a holographic description of M-theory in the light cone frame. In this

model, known as BFSS matrix theory, all spatial dimensions are dynamically generated

while time is put in a prior. Later in [18], another matrix theory is proposed for IIB theory.

This so called IKKT matrix theory is a 0+0 dimensional theory, in contrast to the 0+1

dimensional BFSS theory. Thus in this theory, both spatial and temporal dimensions are

generated dynamically. In fact, the IKKT action is essentially just the D-instanton action.

Thus the D-instanton action provides a holographic description of the full string theory.

We will argue in this note that this also happens for D-instantons in the Misner universe.

One of the advantages of the holographic description is that backreaction can be taken

into account more naturally, since the geometry and objects in it are not treated seperatedly

as in conventional theory including perturbative string theory. And the drawback is that

it is often tricky to get detailed information from these matrices. And in this note we will

encounter both the advantages and disadvantages.

The layout of this note is as follows. In section 2, we review some aspects of the Misner

geometry and properties of closed strings in it. We begin in section 3 the investigation of

D-instanton physics. We derive the matrix action, and identify the vacuum corresponding

to the background geometry. And in section 4, we study how tachyons affect spacetime.

We read from the tachyon deformed D-instanton action the new moduli space, and thus

the resulted geometry. Finally in section 5, we promote our probe action to a second

quantized framework, which can be regarded as an Lorentzian orbifolded version of IKKT

matrix theory. We construct D-branes from matrices and study their properties, giving

evidence that the D-instanton action actually provides a holographic description for Misner

universe.

Recent explorations of other singularities include [40], [41], [42].

Note added: After our paper was submitted to archive, we received another paper [43]

addressing similar problems.

2. Misner universe: The geometry and the closed string story

Misner universe is an orbifold of 1+1-dimensional Minkowski space

ds2 = −2dx+dx− (2.1)

2



by the identification

x+ ∼ e2πγ x+, x− ∼ e−2πγ x−. (2.2)

Coordinate transformation

x+ = T √2

eγθ , x− = T √2

e−γθ (2.3)

can be made to write Misner space as

ds2 = −dT 2 + γ 2T 2dθ2, (2.4)

with θ ∼= θ + 2π. It is easy to see from (2.4) that this space-time contains two cosmological

regions connected by a space-like singularity.

There are generally two kinds of closed strings in Misner universe: twisted and un-

twisted. Untwisted states include in particular the gravitons and their behaviors are

particle-like. Their wave functions can be obtained by superposing a plane wave in the

parent Minkowski space with its images under the boost (2.2), and is written as [5] [8]

f j,m2,s(x+, x−) =

∞−∞

dveip+X−e−2πγv+ip−X+e2πγv+ivj+vs, (2.5)

with j the boost momentum, m the mass, and s the SO(1, 1) spin in R(1,1).

Due to the orbifold projection (2.2), new twisted sectors arise in Misner space with

strings satisfyingX ±(τ, σ + 2π) = e±2πγ wX ±(τ, σ), (2.6)

where the winding number w is an integer. Many mysterieses of the Misner universe have

origin from these winding strings. It was shown in [7] that there exists a delta-function

normalizable continuum of physical twisted states, which can be pair produced in analogy

with the Schwinger effect in an electric field. And evaluating the Bogolubov coefficients,

they showed that the transmission coefficient reads

q4 = e−πM 2

/2ν cosh(π˜

M

2

/2ν )| sinh πj | , (2.7)

where ν = −γw is the product of the boost parameter and the winding number, and

M 2 = α+0 α−0 + α−0 α+

0 , M̃ 2 = α̃+0 α̃−0 + α̃−0 α̃+

0 (2.8)

with string zero modes α±0 and α̃±0 , comes from the Virasoro conditions.

3



3. D-instantons probing Misner universe

We embed the geometry (2.1) (2.4) into string theory by adding another 8 flat direc-

tions Y a, a = 1, . . . , 8, and then put N D-instantons in this geometry then go on to find

the field theory describing their behavior. We want to read from the modular space of the

D-instantons the background geometry, following the study of [11] [19]. In this note we

ignore the backreaction of these D-instantons.

D-brane dynamics on the orbifolds were variously discussed in the previous literature.

We follow mainly Taylor’s procedure [20]. The open string degrees of freedom form a

matrix theory. We focus on the bosonic part, which are the embedding coordinates. Go

to the covering space

(X +, X −) ∈ R1,1, Y a ∈ R8⊥, (3.1)

and make the projection(2.2), then each D-instanton has infinitely many images, whichcan be captured by matrices of infinitely many blocks. Each block is itself a N ×N matrix.

The orbfold projection for these blocks reads

X +i,j = e2πγ X +i−1,j−1,

X −i,j = e−2πγ X −i−1,j−1,

Y ai,j = Y ai−1,j−1.

(3.2)

These matrices can be solved using the following basis:

(β ml )ij = e2πilγ δi,j−m. (3.3)

Some of their communication relations will be used in this note:

[β m0 , β m′

0 ] = 0

[β m0 , β m′

1 ] = (e2πmγ − 1)β m+m′

1

[β m0 , β m′

−1] = (e−2πmγ − 1)β m+m′

−1

[β m1 , β m′

−1] = (e−2πmγ − e2πm′γ )β m+m′

0 .

(3.4)

The solutions thus readX + =

m∈Z

x+mβ m1 ,

X − =

m∈Z

x−mβ m−1,

Y a =

m∈Z

yamβ m0 .

(3.5)

4



The low energy effective action for the D-instantons can be obtained from dimensional

reduction of 10-d Super Yang-Mills, and keep only the bosonic part, we get

S =1

2g2

Z 0

9

µ,ν=0

Tr([X µ, X ν][X µ, X ν ]), (3.6)

with coupling g2 = gsα′2 , where we eliminate factors of order 1; Z 0 is the normalization

factor which is formally trace of the infinite dimensional unite matrix. The above action

is written in the Minkowski signature, so there is an overall sign difference with IKKT[18].

Written in terms of the above solution(3.5), the action reads

S = − 1

g2

m+m′+n+n′=0

Tr

x+mx+m′x

−n x−n′(e−2πmγ − e2πnγ )(e−2πm′γ − e2πn′γ )

+ 2x+mx−m′yanya

n′(e2πnγ − 1)(e2πn′γ − 1)e−2π(m+n)γ

.

(3.7)

The above action has many branches of vacuum. In the following, we will consider the

Higgs branch which corresponds to D-branes probing the Misner part of the geometry,

with the same coordinates in the other 8 directions. Thus we can eliminate the second

term of the above action.

The infinite summation in (3.7) indicates a ”hidden” dimension with topology S 1, on

which the Fourier coefficients of a real scalar field can represent the modes in ( 3.5). That

is

x+m =

2π

0

dσ√2π

X +(σ)e−imσ,

x−m =

2π

0

dσ√2π

X −(σ)e−imσ,

(3.8)

and the action

S = − 1

g2

2π

0

dσ

2πTr

[ei2πγ d

dσ X +(σ)]X −(σ) − [e−i2πγ ddσ X −(σ)]X +(σ)

2

= − 1g2

2π

0

dσ2π

Tr

X +(σ + i2πγ )X −(σ) − X −(σ − i2πγ )X +(σ)2

(3.9)

is complemented by the symmetry

X +(σ) → e2πγ X +(σ),

X −(σ) → e−2πγ X −(σ),(3.10)

5



inherited from the orbifold projection (2.2).

Note that the system (3.9) (3.10) possesses a large variety of vacua, which are simply

solutions of the equation

X +

(σ + i2πγ )X −

(σ) − X −

(σ − i2πγ )X +

(σ) = 0. (3.11)

Just to study the space-time geometry, we take all the D-instantons to coincide, and the

matrices become ordinary functions. We can define

F (σ) ≡ X +(σ + i2πγ )X −(σ), (3.12)

which appears repeatedly in this note. And the vacuum condition reads now

F (σ) = F (σ − i2πγ ). (3.13)

Note also that X +, X − are all defined on a circle, which says

F (σ) = F (σ + 2π). (3.14)

For the problems in hand, we expect X +(σ), X −(σ) to have no poles in the σ plane, so

F (σ) must be a constant.

Thus X +(σ + i2πγ ) can be factorized as a real function of σ with periodicity 2π

multiplied by a constantX +(σ + i2πγ ) = αf (σ). (3.15)

For real functions X +(σ), f (σ) with periodicity 2π, we can expand them as

X +(σ) =+∞

n=−∞

cneinσ, f (σ) =+∞

n=−∞

f neinσ, (3.16)

with c−n = c∗n and f −n = f ∗n. Then eq.(3.15) leads to

cne−2πnγ = αf n, (3.17)

and thus

c−ne2πnγ = αf −n, (3.18)

or

c∗ne2πnγ = αf ∗n, (3.19)

6



and

c∗ne−2πnγ = α∗f ∗n. (3.20)

For any non vanishing cn, f n, eq.(3.19) (3.20) require

e4πnγ = αα∗

, (3.21)

which obviously can not be satisfied for more than one value of n. And note that for n

non-zero, cn, f n are paired with c−n, f −n. So all cn, f n except c0, f 0 must vanish, and thus

X +(σ) must be a constant, which subsequently forces X −(σ) also to be a constant.

Taking into account the constraint (3.10), we get a branch of the moduli space (the

Higgs branch)

M =

X +

, X −

, Y a

∈ RX + ∼= e2πγ X +

X − ∼= e−2πγ X −

(3.22)

which is exactly the original Misner universe.

To end this section, we remind the reader of some characteristics of the action(3.9).

First, it is non-local. And the physical origin is still mysterious to us. At first glance one

may think winding modes can cause such non-locality. But from the above calculation we

see that the effect of these twisted sectors is to induce the infinite summation in eq.(3.7)

and thus only leaving trace in the necessity to use an integral in eq. (3.9). In the null

brane case [21], where there are similarly twisted sector contributions, D-instanton actionis calculated in [19], which is also an integral but with the integrand local. And we see

that the non-locality is very peculiar to Misner space whose singularity is spacelike.

It was shown in [22] by Hashimoto and Sethi that the gauge theory on the D3-branes

in the null brane [21] background is noncommutative, thus also non-local. What is in-

teresting in their model is that they observe that upon taking some decoupling limit, the

noncommutative field theory provides a holographic description of the corresponding time-

dependent closed string background (see also [23]). Whether some decoupling limit [24]

exists in our case is worth exploring.Second, notice that the argument in the action (3.9) is complexified, which is a peculiar

property of some time-dependent backgrounds. And it is also a crucial ingredient in our

following treatment of instability of Misner space and of the branes therein. Complexified

arguments also make appearance in the study of other singularities (see for example [25],

[26]).

7



4. D-instantons probing tachyon deformed Misner universe

We go on to deal with the case with winding string tachyon condensates turned on

[14]. Take anti-periodic boundary conditions around the θ circle in (2.4). In the regime

γ 2T 2 ≤ l2s, (4.1)

some winding closed string modes become tachyonic which signals the instability of the

spacetime itself. These modes grow and deform the spacetime. It was speculated in [14]

that the regime (4.1) will be replaced by a new phase with all closed string excitations

lifted.

D-instantons feel the change in the geometry through its coupling to the metric. It was

shown by Douglas and Moore in [27] that the leading effect of tachyons on the Euclidean

orbifolds is to induce a FI-type term in the D-brane potential. This effect comes from the

disk amplitude with one insertion of the twisted sector tachyon field at the center and two

open string vertex operators at the boundary. With a detailed analysis of the full quiver

gauge theory, which provides a description for D-branes on the orbifolds, they combine

the FI term with the Born-Infeld action and the kinetic energies of the hypermultiplets,

and then integrate out the auxiliary D-fields in the vectormultiplet, to find that the effect

of the twisted sector fields is to add a term in the complete square. In our case, we are

dealing with a Lorentzian orbifold which is more subtle than its Euclidean cousin. But tostudy the D-instanton theory, we can perform a wick rotation to go to the Euclidean case,

where the result of [27] will be consulted, and finally we get schematically

S = − 1

g2

2π

0

dσ

2π

X +(σ + i2πγ )X −(σ) − X −(σ − i2πγ )X +(σ) − U (σ)

2. (4.2)

The detailed form of U (σ) is not important in the following treatment where we require

only the existence of such a non-zero term. There may be some subtlty in the above wick

rotation which deserves further clarification. And the above D-instanton action can also

be thought of as coming from a time-like T-dual [28] of a more controllable system with

D-particles on an Euclidean orbifold.

The vacuum condition becomes now

X +(σ + i2πγ )X −(σ) − X −(σ − i2πγ )X +(σ) = U (σ). (4.3)

8



And the leading effect of the tachyons is to make the geometry noncommutative

[X +(σ), X −(σ)] = U (σ). (4.4)

The above approximation essentially sets γ = 0. And we know that the geometry cor-

responding to γ = 0 is just the flat space without any boost identification, so one may

think this case can not teach us much about Misner space. But we note that the term

U (σ) encodes information peculiar to Misner space. From our experiences for other better

understood tachyons [29] [11] [12], we can think this way: the nontrivial boost identifica-

tion, plus the anti-periodic boundary condition for fermions, first cooks some closed string

tachyons. Then these tachyons condense. For these stages we cannot say anything new

in the above formalism. We intend only to explore how subsequently spacetime geometry

is modified by these tachyon condensates, taking into account the fact that the tachyons

couple to the metric. At this stage, the process is driven by the tachyon source while the

nontrivial boost identification is not essential, thus we can use the above approximation

γ = 0.

To see the picture more clearly, let’s go to the (T, θ) frame. We can model the geometry

by choosing

eγθ T e−γθ = aT, (4.5)

with a some constant. This makes a fuzzy cone, where the deviation of a from unity

measures the fuzziness of the geometry. And the noncommutative relation (4.4) reads now

T 2(a − 1

a) = 2U. (4.6)

We see from the above equation that in the two asymptotic regions T → ±∞, a goes to

unity, thus T commutes with θ and conventional geometric notion works well. But as T

goes to zero, a deviates more and more from unity. Thus spacetime becomes more and

more fuzzy. At the origin T = 0, a diverges, and the conventional notion of geometry

breaks down totally. Thus the original spacelike singularity is removed.

4.1. Comparison with McGreevy and Silverstein’ s Nothing Phase

It is also interesting to compare our result with that of [14] (see also [30]) which

actually motivated our study. They employ perturbative string methods, working on the

world sheet using techniques from Liouville theory. Here we will intend to propose a non-

perturbative formulation of the theory, and the emerging picture is in fact consistent with

9



their work.∗∗ They read from their 1-loop partition function that the volume of the time

direction is truncated to the region without closed string tachyon condensates, providing

evidence for previously expected picture that closed string tachyon condensation lifts all

closed string degrees of freedom, leaving behind a phase of ”Nothing”. In our formalism,

we can say more about this ”Nothing Phase”. Although ordinary concepts of spacetime

break down, we can still model such region by some non-commutative geometry. Although

closed string degrees of freedom cease to exist in such region, it is nevertheless possible to

formulate the theory with open string degrees of freedom. And we expect the D-instanton

matrix action (4.2) can serve this role. It seems that matrix models have the potential to

say more about closed string tachyons, who are known as killers of closed string degrees of

freedom, as open string tachyons did for open string degress of freedom.

Recently it was also found [40] that near some null singularities, the usual supergravityand even the perturbative string theory break down. Matrix degrees of freedom become

essential and the theory is more suitably described by a Matrix string theory. Such non-

abelian behavior seems intrinsic for singularities.

4.2. A model for big crunch/big bang transition

The whole picture of the resulted spacetime after tachyon condensation is that of two

asymptotically flat region, the infinite past and infinite future, connected by some fuzzy

cone. And although conventional concept of time breaks down, there is still causal connec-

tion between the infinite past and infinite future. This fact is cosmologically attractive.

An alternative to inflation is proposed in [31], where they considered the possibility

that the big bang singularity is not the termination of time, but a transition from the

contracting big crunch phase to the expanding big bang phase. The horizon problem is

nullified in this scenario, and other cosmological puzzles may also be solved in this new

framework. Unfortunately it is generally difficult to get a controllable model for such a

scenario. From the above discussion, we see that the tachyon deformed Misner universeserves as a concrete model for such big crunch/big bang transition [31].

**We give literally different answer to the question: can time start or end by turning on such

closed string tachyons, where we employ different interpretation of the question. They say yes

[14] where they mean conventional aspect of time breaks down in some region. And we say no

having in mind that information can still be transferred from infinite past to infinite future.

10



4.3. Remarks on space-time noncommutativity

Note that the fuzzy cone condition (4.5) is just a statement that space and time do not

commute near the spacelike singularity. This leads to the stringy spacetime uncertainty

relation, which was suggested by Yoneya and Li to be a universal characterization of short

distance structure for string and D-brane physics [32]. Here their idea is realized in a time-

dependent background and we can compare with Hashimoto and Sethi’s realization [22] of

time-dependent space-space noncommutativity. In their case, the noncommutativity can

be traced back to the presence of background B field, which is well understood [33]. For

Misner space the noncommutativity all comes from the violent fluctuations of the geometry

near the singularity, and this needs further study.

Space-time noncommutativity is also interesting for cosmology, since it leads to cou-

pling between inflation induced fluctuations and the background cosmology thus may pro-

duce transplackian effects. This subject is much explored in the literature [ 34], where it was

pointed out that short distance dispersion relations may be modified, and non-Gaussianity,

anisotropism and the running of spectual index can be explained. But usually for lack of

concrete models, discussions are made generally. In the model of tachyon deformed Misner

universe, more detailed questions can be asked.

5. A possible holographic description of Misner universe

Now we ask the question: how much information about the Misner space is encoded

in the action (3.9)? In fact, the original action (3.6) plus the fermionic part and a chemical

potential term proportional to Tr1 is proposed in [18] to provide a constructive definition

of definition of type IIB string theory. This so called IKKT matrix theory, is interpreted

in [35] as the D-instanton counterpart of the D0-brane matrix theory of BFSS[17]. Fun-

damental strings and Dp-branes [18] [35] [36] can be constructed from such matrices, and

long-distance interaction potentials of BPS configurations computed from such matrices

match the supergravity results [37]. We can extend this matrix/string correspondence

to the Misner case by making orbifold projections (2.2) on both sides, where obviously

the projection commutes with the matrix/string mapping. So we propose that the ac-

tion (3.9) (plus its fermionic counterpart and possible chemical potential term) provides a

holographic description of the Misner universe.

For these 0+0 (or here under orbifold projection, 1+0) dimensional model, time is not

put in a priori, but generated dynamically. This may be the underlying reason why our

11



description of change of the structure of time is possible. And it also indicates that these

matrix models may have privilege in the description of spacelike singularities and other

time-dependent systems.

We go on to construct branes in Misner universe. The equation of motion of action

(3.6) is

gµρgνσ [X ν , [X ρ, X σ]] = 0, (5.1)

which in components are

[X −, [X −, X +]] − [Y a, [X −, Y a]] = 0,

[X +, [X +, X −]] − [Y a, [X +, Y a]] = 0,

[X +, [Y a, X −]] + [X −, [Y a, X +]] = 0.

(5.2)

As above we expand the matrices in the β basis (3.3), and make the transformation (3.8),

and choose Y (σ) to constant. Thus we get the classical solutions

dσ

2πTrX −(σ)

X +(σ + i4πγ )X −(σ + i2πγ ) − X −(σ)X +(σ + i2πγ )

− X +(σ + i2πγ )X −(σ) + X −(σ − i2πγ )X +(σ)

= 0,

dσ

2πTrX +(σ)

X −(σ − i4πγ )X +(σ − i2πγ ) − X +(σ)X −(σ − i2πγ )

− X −(σ − i2πγ )X +(σ) + X +(σ + i2πγ )X −(σ)

= 0.

(5.3)

which are unusually integral equations.

We can define

L(σ) ≡ X +(σ + i2πγ )X −(σ) − X −(σ − i2πγ )X +(σ), (5.4)

which appears repeatedly in this note. Note that L(σ) = 0 is just the vacuum (3.11).

Consider in the moduli space (3.22) a special configuration

X +(σ) = X −(σ) =

t1t2

. . .

tN

,

Y a = 0,

(5.5)

12



where t(i) ’s are constants. In the large N limit, we see from the corresponding classical

trajectory

T = t, θ = 0, Y a = 0 (5.6)

with parameter t, that it is just a D0-brane[18]. And the D-instanton action at this point

of the moduli space reproduces the D0-brane action of the BFSS matrix theory [18].

It seems strange that D0-branes emerge this way from a D-instanton matrix model

which is directly related to the type IIB theory. In flat space, these D0-branes are super-

symmetric and stable.∗ And in Misner space they will also not decay for their seemingly

wrong dimension. The IKKT proposal [18] is that since type IIA string theory is related to

type IIB theory by T-duality, in some regions of the type IIB moduli space, the type IIA

theory can emerge as a more suitable description. And the existence of these D0-branes

will be considered as manifestation of duality.

It is pointed out in [16] that D0-branes in Misner universe are actually unstable, they

are subject to open string pair creation. Since we regard our instanton action to be a

second quantized description of Misner universe, such phenomena should be reproduced.

Let’s make some small perturbation around the D0-brane (5.5)

X +(σ) =

t1 + δt1t2

. . .

tN

,

X −(σ) =

t1t2

. . .

tN

,

Y a = 0,

(5.7)

with δt(1) = ǫσ. Now the action becomes

S = − 1

g2 dσ

2πt21[δt1(σ + i2πγ ) − δt1(σ)]2

= (2πγ t1

g)2ǫ2.

(5.8)

* Brane charges are more subtle in the IKKT matrix theory than that of BFSS [39]. Some

proposals were made in [36] for Dp-branes with p odd. But for p even, (linear combinations of )

the matrices commute with each other, leaving no room for constructing central charges along the

lines of [39].

13



Note the sign change above, which originates from the complexified arguments in the

integrand. And perturbations of other eigenvalues give similar results.

To understand the above action, consider a quantum mechanical system

S =

dt 1

2˙

X 2

− U (X )

. (5.9)

With U (X ) = −12kX 2 and k > 0, it is just a particle moving in an inverted harmonic

potential. Seemingly the particle can not stay static. It will roll down the potential.

When the potential term dominates the whole action, we go over to the action (5.8). And

accordingly the D0-branes are unstable. Worse still, the action is even not bounded from

below, which makes it impossible to define a first quantized vacuum. This fact has already

been noticed in [7] in their study of perturbative string theory of Misner universe.

Such kind of inverted harmonic potential also appears in c = 1 matrix model, and there

closed string emission from unstable D0-branes is described by a matrix eigenvalue rolling

down such a potential [38]. In our formalism, description of such dynamical processes is

intrinsically subtle, where technically the difficulty stems from the fact that we do not have

kinetic terms for the matrix eigenvalues. But we can understand from the path integral

point of view that, the smaller the Euclidean action S E = −S , the more the configuration

contributes to the whole amplitude. And if we start with the D0-brane (5.5), quantum

fluctuations will generally destroy this configuration, driving the system to more probable

configurations with larger ti, making the brane effectively ”roll down” the potential. In

the large N limit, this corresponds to the phenomena that the unstable D0-branes emit

closed strings and/or open string pairs [38].

For the background geometry

X +(σ) = x+I N ×N ,

X −(σ) = x−I N ×N ,

Y a(σ) = yaI N ×N,

(5.10)

we can likely make a perturbation

X +(σ) =

x+ + δx+

x+

. . .

x+

,

X −(σ) = x−I N ×N ,

Y a(σ) = yaI N ×N .

(5.11)

14



And similarly we get an inverted harmonic potential with δx+ a linear perturbation, thus

the same instability arises, which is consistent with what is found in perturbative string

theory in Misner space [7] [8] which states that the vacua is unstable while winding strings

are pair produced as a consequence of the singular geometry in analogy with the Schwinger

effect in an external electric field. This is a tunnelling process, matching precisely our de-

scription via D-instantons. And in this matrix framework, we can see that the instabilities

of the geometry and the branes have essentially the same origin. Both can be interpreted

as matrix eigenvalues ”rolling down” a unbounded-from-below potential.

Next let’s discuss the D-strings. It is easy to see from (5.3) that L(σ) = constant

is a classical solution. The Minkowski limit γ → 0 of L(σ) is just the commutator

[X +(σ), X −(σ)], and in this limit L(σ) = constant becomes the familiar result in ma-

trix theory

[X +(σ), X −

(σ)] = iF +−I N ×N , (5.12)

with F +− some non-zero constant. And there in the large N limit, it represents D-strings

[18] or some non-marginal bound states of D-strings with D-instantons [35].

Here the solution corresponding to a D-string is

X +(σ) =L+

√2πN

q,

X −(σ) =L−√2πN

p,

Y a = 0,

(5.13)

with L+, L− some large enough compactification radius, and the N ×N hermitian matrices

0 ≤ q, p ≤ √2πN satisfying

[q, p] = I N ×N , (5.14)

which is obviously only valid at large N . Note also the omitted i in our convention in

contrast to usual notion.

These D-strings are also unstable [16], and the interpretation in matrix theory is

essentially the same as for D0-branes and the geometry. We add some small perturbations,say change q11 to q11 + ǫσ, and the real part of the action becomes now

S pert = S D1 + (2πγp11

g)2ǫ2, (5.15)

leading to the ”rolling” behavior of the matrix elements and thus D-string’s emitting open

or closed strings.

15



The universality of the interpretation of instabilites of D0- and D1-branes provides

further evidence that D0-branes do not decay for their ”wrong dimensionality” and the

region around (5.5) has a more suitable description as type IIA string theory.

Obviously more efforts are needed to figure out the details of the string emission, such

as the spectrum and emission rate which have already been calculated in perturbative

string theory [8] [16]. The matrix formalism has the potential advantage to treat more

precisely the backreaction of the emitted strings as we have exampled in section 4.

Acknowledgments

I thank Bin Chen, Qing-Guo Huang, Miao Li and Peng Zhang for valuable discussions.

And in particular it’s my pleasure to thank Miao Li for permanent support and insightful

comments at different stages of this work.

16



References

[1] Lance J. Dixon, Jeffrey A. Harvey, C. Vafa, Edward Witten, ”Strings on orbifolds”,

Nucl.Phys.B261:678-686,1985; ”Strings on orbifolds.2”, Nucl.Phys.B274:285-314,1986.

[2] A. Strominger, ”Massless black holes and conifolds in string theory”, Nucl. phys. B

451, 96 (1995), hep-th/9504090.

[3] Clifford V. Johnson, Amanda W. Peet, Joseph Polchinski, ”Gauge Theory and the

Excision of Repulson Singularities”, Phys.Rev. D61 (2000) 086001, hep-th/9911161.

[4] Gary T. Horowitz, Alan R. Steif, ”Singular string solution with nonsigular initial

data”, Phys.Lett.B258:91-96,1991.

[5] Nikita A. Nekrasov, ”Milne universe, tachyons and quantum group”, Surveys High

Energ.Phys.17:115-124,2002, e-Print Archive: hep-th/0203112.

[6] Micha Berkooz, Ben Craps, David Kutasov, Govindan Rajesh, ”Comments on cos-

mological singularities in string theory”, JHEP 0303:031,2003, e-Print Archive: hep-

th/0212215.

[7] M. Berkooz, B. Pioline, ”Strings in an electric field, and the Milne Universe”, JCAP

0311 (2003) 007, hep-th/0307280.

[8] M. Berkooz, B. Pioline, M. Rozali, ”Closed Strings in Misner Space: Cosmological

Production of Winding Strings”, JCAP 0408 (2004) 004, hep-th/0405126.

[9] M. Berkooz, B. Durin, B. Pioline, D. Reichmann, ”Closed Strings in Misner Space:

Stringy Fuzziness with a Twist”, JCAP 0410 (2004) 002, hep-th/0407216.

[10] Bruno Durin, Boris Pioline, ”Closed strings in Misner space: a toy model for a Big

Bounce ?”, Proceedings of the NATO ASI and EC Summer School “String The-

ory: from Gauge Interactions to Cosmology”, Cargese, France, June 7-19, 2004, hep-th/0501145.

[11] A. Adams, J. Polchinski, E. Silverstein, ”Don’t panic! Closed string tachyons in ALE

space-times”, JHEP 0110:029,2001, e-Print Archive: hep-th/0108075

[12] Matthew Headrick, Shiraz Minwalla, Tadashi Takayanagi, ”Closed String Tachyon

Condensation: An Overview”, Class.Quant.Grav. 21 (2004) S1539-S1565, e-Print

Archive:hep-th/0405064.

[13] A. Adams, X. Liu, J. McGreevy, A. Saltman, E. Silverstein, ”Things Fall Apart:

Topology Change from Winding Tachyons”, hep-th/0502021.

[14] John McGreevy, Eva Silverstein, ”The Tachyon at the End of the Universe”,hep-th/0506130.

[15] Michael R. Douglas, Daniel Kabat, Philippe Pouliot, Stephen H. Shenker, ”D-

branes and Short Distances in String Theory”, Nucl.Phys. B485 (1997) 85-127, hep-

th/9608024.

[16] Yasuaki Hikida, Rashmi R. Nayak, Kamal L. Panigrahi, ”D-branes in a Big Bang/Big

Crunch Universe: Misner Space”, hep-th/0508003.

17

http://arxiv.org/abs/hep-th/9504090






































[17] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, ”M Theory As A Matrix Model: A

Conjecture”, Phys.Rev. D55 (1997) 5112-5128, hep-th/9610043.

[18] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, ”A Large-N Reduced Model as

Superstring”, Nucl.Phys. B498 (1997) 467, hep-th/9612115.

[19] Micha Berkooz, Zohar Komargodski, Dori Reichmann, Vadim Shpitalnik, ”Flow of geometry and instantons on the null orbifold”, hep-th/0507067.

[20] Washington Taylor, IV,”D-brane field theory on compact spaces”, Phys.Lett.B394:283-

287,1997, e-Print Archive: hep-th/9611042.

[21] Jos Figueroa-O’Farrill, Joan Simn, ”Generalised supersymmetric fluxbranes”, JHEP

0112 (2001) 011, hep-th/0110170.

[22] Akikazu Hashimoto, Savdeep Sethi, ”Holography and string dynamics in time depen-

dent backgrouns”, Phys.Rev.Lett.89:261601,2002, e-Print Archive: hep-th/0208126.

[23] Joan Simn, ”Null orbifolds in AdS, Time Dependence and Holography”, JHEP 0210

(2002) 036, hep-th/0208165; Rong-Gen Cai, Jian-Xin Lu, Nobuyoshi Ohta, ”NCOS

and D-branes in Time-dependent Backgrounds”, Phys.Lett. B551 (2003) 178-186, hep-

th/0210206.

[24] Hirosi Ooguri, Kostas Skenderis, ”On The Field Theory Limit Of D-Instantons”, JHEP

9811 (1998) 013, hep-th/9810128.

[25] Lukasz Fidkowski, Veronika Hubeny, Matthew Kleban, Stephen Shenker, ”The Black

Hole Singularity in AdS/CFT”, JHEP 0402 (2004) 014, hep-th/0306170.

[26] Guido Festuccia, Hong Liu, ”Excursions beyond the horizon: Black hole singularities

in Yang-Mills theories (I)”, hep-th/0506202.

[27] Michael R. Douglas, Gregory Moore, ”D-branes, Quivers, and ALE Instantons”, hep-

th/9603167.[28] C.M. Hull, ”Timelike T-Duality, de Sitter Space, Large N Gauge Theories and Topo-

logical Field Theory”, JHEP 9807 (1998) 021, hep-th/9806146; ”Duality and the Sig-

nature of Space-Time”, JHEP 9811 (1998) 017, hep-th/9807127.

[29] Ashoke Sen, ”Tachyon Condensation on the Brane Antibrane System”, JHEP

9808 (1998) 012, hep-th/9805170; ”Rolling Tachyon”, JHEP 0204 (2002) 048, hep-

th/0203211; ”Tachyon Dynamics in Open String Theory”, Int.J.Mod.Phys. A20 (2005)

5513-5656, hep-th/0410103.

[30] Eva silverstein, ”The tachyon at the end of the universe”, talk at string2005,

http://www.fields.utoronto.ca/audio/05-06/strings/silverstein/.[31] Justin Khoury, Burt A. Ovrut, Nathan Seiberg, Paul J. Steinhardt, Neil Turok, ”From

Big Crunch to Big Bang”, Phys.Rev. D65 (2002) 086007, hep-th/0108187.

[32] Y. Yoneya, in ”Wandering in the Fields”, eds. K. Kawarabayashi, A. Ukawa

(World Scientific, 1987), p419; Miao Li, Tamiaki Yoneya, ”D-Particle Dynamics and

The Space-Time Uncertainty Relation”, Phys.Rev.Lett. 78 (1997) 1219-1222, hep-

th/9611072; Miao Li, Tamiaki Yoneya, ”Short-Distance Space-Time Structure and

18





















http://www.fields.utoronto.ca/audio/05-06/strings/silverstein/







http://www.fields.utoronto.ca/audio/05-06/strings/silverstein/























Black Holes in String Theory : A Short Review of the Present Status”, in the spe-

cial issue of the Journal of Chaos, Solitons and Fractals on ”Superstrings, M, F,

S...Theory”, hep-th/9806240; Tamiaki Yoneya, ”String Theory and the Space-Time

Uncertainty Principle”, Prog.Theor.Phys. 103 (2000) 1081-1125, hep-th/0004074.

[33] Nathan Seiberg, Edward Witten, ”String Theory and Noncommutative Geometry”,JHEP 9909 (1999) 032, hep-th/9908142.

[34] Chong-Sun Chu, Brian R. Greene, Gary Shiu, ”Remarks on Inflation and Noncommu-

tative Geometry”, Mod.Phys.Lett. A16 (2001) 2231-2240, hep-th/0011241; Stephon

Alexander, Robert Brandenberger, Joao Magueijo, ”Non-Commutative Inflation”,

Phys.Rev. D67 (2003) 081301, hep-th/0108190; Robert Brandenberger, Pei-Ming Ho,

”Noncommutative Spacetime, Stringy Spacetime Uncertainty Principle, and Density

Fluctuations”, Phys.Rev. D66 (2002) 023517; AAPPS Bull. 12N1 (2002) 10-20, hep-

th/0203119; Qing Guo Huang, Miao Li, ”CMB Power Spectrum from Noncommu-

tative Spacetime”, JHEP 0306 (2003) 014, hep-th/0304203; Qing-Guo Huang, Miao

Li, ”Noncommutative Inflation and the CMB Multipoles”, JCAP 0311 (2003) 001,

astro-ph/0308458; Qing-Guo Huang, Miao Li, ”Power Spectra in Spacetime Noncom-

mutative Inflation”, Nucl.Phys. B713 (2005) 219-234, astro-ph/0311378.

[35] A.A. Tseytlin, ”On non-abelian generalisation of Born-Infeld action in string theory”,

Nucl.Phys. B501 (1997) 41-52, hep-th/9701125.

[36] Miao Li, ”Strings from IIB Matrices”, Nucl.Phys.B499(1997)149-158, hep-th/9612222;

I. Chepelev, Y. Makeenko, K. Zarembo, ”Properties of D-Branes in Matrix Model of

IIB Superstring”, Phys.Lett. B400 (1997) 43-51, hep-th/9701151; Ansar Fayyazuddin,

Douglas J. Smith, ”P-brane solutions in IKKT IIB matrix theory”, Mod.Phys.Lett.

A12 (1997) 1447-1454; hep-th/9701168; A. Fayyazuddin, Y. Makeenko, P. Olesen, D.J.Smith, K. Zarembo, ”Towards a Non-perturbative Formulation of IIB Superstrings by

Matrix Models”, Nucl.Phys. B499 (1997) 159-182, hep-th/9703038.

[37] I. Chepelev, A.A. Tseytlin, ”Interactions of type IIB D-branes from D-instanton ma-

trix model”, Nucl.Phys. B511 (1998) 629-646, hep-th/9705120.

[38] John McGreevy, Herman Verlinde, ”Strings from Tachyons”, JHEP 0312 (2003) 054,

hep-th/0304224; Igor R. Klebanov, Juan Maldacena, Nathan Seiberg, ”D-brane Decay

in Two-Dimensional String Theory”, JHEP 0307 (2003) 045, hep-th/0305159.

[39] Tom Banks, Nathan Seiberg, Stephen Shenker, ”Branes from Matrices”, Nucl.Phys.

B490 (1997) 91-106, hep-th/9612157.[40] Ben Craps, Savdeep Sethi, Erik Verlinde,”A Matrix Big Bang”, hep-th/0506180; Miao

Li, ”A Class of Cosmological Matrix Models”, hep-th/0506260; Miao Li, Wei Song,

”Shock Waves and Cosmological Matrix Models”, hep-th/0507185; Sumit R. Das,

Jeremy Michelson, ”pp Wave Big Bangs: Matrix Strings and Shrinking Fuzzy Spheres”,

hep-th/0508068; Bin Chen, ”The Time-dependent Supersymmetric Configurations in

M-theory and Matrix Models”, hep-th/0508191; Bin Chen, Ya-li He, Peng Zhang, ”

19









http://arxiv.org/abs/astro-ph/0308458










































Exactly Solvable Model of Superstring in Plane-wave Background with Linear Null

Dilaton”, hep-th/0509113.

[41] Thomas Hertog, Gary T. Horowitz, ”Holographic Description of AdS Cosmologies”,

JHEP 0504 (2005) 005, hep-th/0503071.

[42] Haitang Yang, Barton Zwiebach, ”Rolling Closed String Tachyons and the BigCrunch”, hep-th/0506076; Haitang Yang, Barton Zwiebach, ”A Closed String Tachyon

Vacuum ?”, hep-th/0506077.

[43] Yasuaki Hikida, Ta-Sheng Tai, ”D-instantons and Closed String Tachyons in Misner

Space”, hep-th/0510129.

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Jian-Huang She- A Matrix Model for Misner Universe and Closed String Tachyons

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Transcript of Jian-Huang She- A Matrix Model for Misner Universe and Closed String Tachyons